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Kinetics of first-order phase transitionsfrom microcanonical thermostatistics
L. G. Rizzi11Departamento de Fısica, Universidade Federal de Vicosa,
Av. P. H. Rolfs, s/n, 36570-900, Vicosa-MG, Brazil.
E-mail: [email protected]
1 July 2020
Abstract. More than a century has passed since van’t Hoff and Arrhenius formulated
their celebrated rate theories, but there are still elusive aspects in the temperature-
dependent phase transition kinetics of molecular systems. Here I present a theory based
on microcanonical thermostatistics that establishes a simple and direct temperature
dependence for all rate constants, including the forward and the reverse rate constants,
the equilibrium constant, and the nucleation rate. By considering a generic model
that mimic the microcanonical temperature of molecular systems in a region close
to a first-order phase transition, I obtain shape-free relations between kinetics
and thermodynamics physical quantities which are validated through stochastic
simulations. Additionally, the rate theory is applied to results obtained from protein
folding and ice nucleation experiments, demonstrating that the expressions derived here
can be used to describe the experimental data of a wide range of molecular systems.
Keywords: first-order phase transitions, microcanonical thermostatistics, molecular
systems, rate constants, nucleation rate.
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Contents
1 Introduction 2
2 Microcanonical thermostatistics 4
3 First-order phase transition kinetics 6
3.1 Rate constants and the equilibrium rate . . . . . . . . . . . . . . . . . . 7
3.2 Protein folding kinetics and protein stability . . . . . . . . . . . . . . . . 10
3.3 Ice nucleation in supercooled water droplets . . . . . . . . . . . . . . . . 12
4 Concluding remarks 15
Appendix A Microcanonical model for phase transitions 16
Appendix B Stochastic simulations 17
Appendix C Mean first-passage times 19
Appendix D Relaxation rate and the nucleation rate coefficient 19
References 21
1. Introduction
The temperature dependence of first-order phase transitions kinetics in molecular
systems is probably one of the oldest unsolved issues in modern science,
despite its importance to a wide range of processes in biology, climate, and
materials science. Examples include the aggregation of misfolded or intrinsically
disordered proteins, which is a phenomenon that can be related to a number of
proteinopathies [1] (e.g., Alzheimer’s disease and type 2 diabetes); and the formation of
protein crystals, which are used by crystallographic methods in the characterization of
their tridimensional structures [2].
Misunderstandings may have arisen partially due to many Arrhenius-like
expressions presented in the literature which are often used indiscriminately beyond
their scope [3]. It is common to find studies where the rates are defined by expressions
Kinetics of first-order phase transitions from microcanonical thermostatistics 3
proportional to e−∆G‡/kBT with effective activation energies ∆G‡ that are independent of
the temperature T , just as those derived by Eyring [4] and Kramers [5] which underpin
the well-known transition state theories [3]. The problem is that, if ∆G‡ does not display
any dependence on T , the corresponding rate will increase with temperature, however
this is exactly the opposite behaviour observed in processes which take place in many
finite molecular systems, e.g., folding of heteropolymers and molecular crystallization.
Remarkably, “anomalous” behaviours like that are known for decades, but it seems that
they are still causing confusion today [6].
Of particular interest to first-order phase transitions is the temperature-dependent
kinetics of nucleation processes, which is characterized by the nucleation rate j(T ).
Usually, the nucleation rate is defined by an Arrhenius-like expression as
j(T ) = Ae−∆G∗/kBT , (1)
where ∆G∗ can be thought as an energetic barrier that is needed to be overcomed
through fluctuations in order to convert part of the system to the new phase and is
related to work to form the critical nucleus [7, 8]; and A is a pre-factor that determines
the dimensional units of the nucleation rate and is related to the attachment frequencies
to the critical nucleus and to the Zeldovich factor [7]. Equation (1) was first introduced
by Volmer & Weber and, together with the seminal contributions made by Becker &
Doring, Farkas, Frenkel, Gibbs, Kaischew, Stranski, Tammann, and Zeldovich, it forms
the basis of what we know as the classical nucleation theory (CNT) [7]. Unlike the
aforementioned Arrhenius-like expressions related to the rate constants, there were
several attempts to include the temperature dependence on A and ∆G∗ defined in
equation (1). Although successful approaches indicate that ∆G∗ should increase as T
approaches the transition temperature [7, 8], quantitative agreement to experimental
data is rarely observed.
A fundamental drawback of nucleation theories that are based on CNT and
which attempt to describe first-order phase transition kinetics in molecular systems
is their need for assumptions based on geometric features of the system, e.g., that the
critical nucleus should have a spherical shape. References [9, 10] show that this type
of assumption is particularly problematic, specially for systems with anisotropically
interacting molecules.
Interestingly, many years ago, Schmelzer et al. [8] have suggested that an approach
based on microcanonical thermostatistics [11] could have provided an alternative
nucleation theory. Nevertheless, only recently that idea has gained attention with
the works of Janke and collaborators. For instance, Zierenberg et al. [12] explored the
shape-free properties of microcanonical free-energy profiles to describe the aggregation
of polymeric chains without making any assumptions about the geometry of the critical
nuclei. A few years before, a similar approach based on microcanonical thermostatistics
was used to infer the kinetics of polymeric chains close to their folding-unfolding
transitions [13]. As it happens, the authors of references [12, 13] (including myself)
missed the opportunity to identify any temperature dependence on the effective
Kinetics of first-order phase transitions from microcanonical thermostatistics 4
activation energies that were evaluated from the microcanonical free-energy profiles.
In this paper I recall that idea in order to develop a rate theory that is based
on the microcanonical thermostatistics [11]. By doing so, I am able to establish a
straightforward relationship between the thermodynamics and the kinetics of first-order
phase transitions in molecular systems. In particular, I derive simple temperature-
dependent expressions for all rates, including the nucleation rate, which are validated by
both numerical simulations and experimental data on protein folding and ice nucleation
phenomena.
2. Microcanonical thermostatistics
First, I clarify that the rate theory introduced here is different from both microcanonical
transition state [14] and adiabatic nucleation [15] theories. Also, I emphasize that,
although the textbook meaning of microcanonical is that the system have a constant
energy and is completely isolated, here it is used mainly because the approach is based
on the density of states, which is a quantity that describes fundamental properties
of the system regardless its coupling to a thermal reservoir. In fact, as my aim is to
study temperature-driven first-order phase transitions, the terminology microcanonical
thermostatistics is used only to indicate that I am interested in intrinsically finite
systems [11, 16], e.g., molecular systems either with aggregating molecules or with
folding heteropolymers. In that case, one should consider that the molecular system
have a fixed number of molecules N and a constant volume V , so that its microcanonical
entropy is given by
S(E) = kB ln Ω(E) + C , (2)
where Ω(E) is the density of states, i.e., the number of microscopic configurations
which have energy between E and E + ε; kB is the Boltzmann’s constant; and C is
an arbitrary constant which depends on the bin size ε and on a reference value for the
entropy. Importantly, the microcanonical analysis is usually restricted to cases where
E denotes the internal energy of the system, i.e., the sum of potential and kinetic
energies of the molecules in the system (eventually excluding the energy of explicit
solvent molecules or particles that are in a reservoir), just as those defined in molecular
dynamics simulations [17]. One should note that, because N and V are fixed, changes
in the internal energy E and enthalpy H would be equivalent, and variations in the
Helmholtz free energy F would be equal to changes in the Gibbs potential G. Also,
just for the moment, I assume that kB = 1 in order to display the numerical results
in units comparable to those usually obtained from simulations which explore the
microcanonical thermostatistics as an analysis method. It is worth mentioning that the
microcanonical entropy S(E) can be conveniently obtained by several computational
algorithms, e.g., multicanonical [18], broad histogram [19], Wang-Landau [20], and
statistical temperature weighted histogram method [21, 22].
Kinetics of first-order phase transitions from microcanonical thermostatistics 5
One important intensive quantity that can be obtained from the microcanonical
entropy is the inverse of the microcanonical temperature, which is given by
b(E) =dS(E)
dE. (3)
Just like the density of states Ω(E), both b(E) and the microcanonical temperature
T (E) = 1/b(E) can be used to describe fundamental properties of the system and
are uniquely determined by N , V , and E. Regardless if the system display, e.g.,
homogeneous or heterogeneous nucleation, or polymeric folding and/or aggregation
phenomena, first-order phase transitions are generically characterized by the presence
200 300 400 500 600 700
E
1.1
1.15
1.2
1.25
1.3
b (
E )
β *
E-
E+
200 400 600 800 1000 1200 1400
E
400
800
1200
S (
E )
= l
n Ω
(E )
S *(E )
∆S
200 400 600E
0.8
0.9
1.0
T (
E )
T *
200 400 600E
1
2
β *∆
F (
E )
β *∆F
∆Ε = Ε+ − Ε
−
a
b
Figure 1. Microcanonical thermostatistics. (a) Displays the inverse of the
microcanonical temperature b(E) which is characteristic of molecular systems that
present first-order phase transitions (see Appendix A for details). Dashed (red) line
indicate a Maxwell-like construction used to determine the inverse of the transition
temperature β∗ = 1.1489. Inset: microcanonical temperature T (E) = 1/b(E) obtained
by equation (A.1), which present a transition temperature equal to T ∗ = 1/β∗ = 0.8704
(here it is assumed that kB = 1). (b) Continuous (black) and dashed (violet) lines
correspond to the microcanonical entropy S(E) and its linear approximation (at the
transition), S∗(E) = β∗E − β∗F (E−), with F (E−) = E− − T ∗S(E−), respectively.
Inset: microcanonical free-energy profile β∗∆F (E) with minima at energies E− = 232
and E+ = 520, which are indicated by the vertical (grey) lines. Blue arrows indicate
the free-energy barrier β∗∆F † = 1.16 at the energy E∗ = 372, the microcanonical
latent heat, ∆E† = 288, and the microcanonical entropy difference, ∆S† = 331.
Kinetics of first-order phase transitions from microcanonical thermostatistics 6
of a S-shaped curve in the inverse of the microcanonical temperature [11, 16], as showed
in figure 1(a). In order to explore the generic aspects of my approach I evaluate both
b(E) and S(E) from a model function that mimic the microcanonical temperature T (E)
of molecular systems near a first-order phase transition (see Appendix A).
As shown in figure 1(a), the transition temperature T ∗ = 1/β∗ can be estimated
from b(E) through a Maxwell-like construction [12, 16]. Equivalently, the inverse of the
transition temperature can be determined as
β∗ =∆S†
∆E†, (4)
where ∆E† = E+ − E− denotes the energy difference, or the microcanonical latent
heat, between the phases at energies E+ and E−, while ∆S† = S(E+) − S(E−) is the
microcanonical entropy difference, as illustrated in figure 1(b). At the transition, one
can consider β∗F (E) = β∗E − S(E), so that the microcanonical free-energy profile is
defined as
β∗∆F (E) = β∗[F (E)− F (E−)] = S∗(E)− S(E) , (5)
with S∗(E) = β∗(E − E−) + S(E−). Although the linear function S∗(E) looks very
similar to the microcanonical entropy in figure 1(b), there is a convex intruder region
in S(E) between the energies E− and E+, which is evidenced by the free-energy profile
showed in the inset of figure 1(b). Importantly, the microcanonical free-energy profile
given by equation (5) can be directly related to the equal height canonical probability
distribution at β∗ [23, 24], i.e., p∗(E) ∝ e−β∗∆F (E).
3. First-order phase transition kinetics
In order to study the kinetics of the system, I devised a stochastic protocol in Appendix
B that allows one to simulate the interaction between the system and a canonical thermal
reservoir at a constant temperature T , so that the stationary distribution is equal to
the canonical equilibrium distribution at β = 1/kBT (see figure B1), that is,
p(E) =Ω(E) e−βE
Z(β), (6)
where one can evaluate the canonical partition function Z(β) =∑E Ω(E) e−βE by
assuming that the density of states is given by Ω(E) = eS(E), with S(E) obtained
from the inverse of the microcanonical temperature b(E) as discussed in Appendix A.
In order to perform the simulations, I consider that the energy exchange between the
system and the thermal reservoir occur at random, just as in the stochastic processes
defined in reference [25], which are known to be applicable even for non-equilibrium
processes (see Appendix B for details). Also, because the rate constants are evaluated
directly from the mean first-passage times, the simulations are constructed in a way
that the diffusion in the energy space allows one to reconstruct the time variable just
as discussed in reference [26].
Kinetics of first-order phase transitions from microcanonical thermostatistics 7
3.1. Rate constants and the equilibrium rate
Figure 2(a) shows Arrhenius plots for the forward (κ−) and the reverse (κ+) rate
constants, which are obtained respectively from the inverse of the mean first-passage
times [27], τ− and τ+, that were numerically evaluated from the stochastic simulations
through the method of labelled walkers mentioned in reference [28] (see figure B1
in Appendix B for details). Accordingly, as the temperature T = 1/β decreases, the
forward rate constant κ− = 1/τ− increases, which means that the mean first-passage
time τ− for the system to go from E+ to E− decreases with the temperature, so that
more of those events will occur per unity of time. The result in figure 2(b) show that,
although κ− (and κ+) display a behaviour that is typical of non-Arrhenius kinetics,
the logarithmic of the equilibrium constant, κeq = κ−/κ+, might display a remarkably
linear behaviour as function of β. By considering the fit to the numerical data of the
expression obtained from the van’t Hoff equation, that is,
κeq = exp[β∆E −∆S
], (7)
one finds ∆E = 266 and ∆S = 306, so that T ∗ = (β∗)−1 ≈ (∆S/∆E)−1 = 0.87.
Interestingly, as suggested by the analytical expressions derived in the following, the
1.12 1.14 1.16 1.18
β = 1/T
-18
-16
-14
-12
-10
-8
rate
co
nst
ants
κ+
κ- lnκ
−
lnκ+
Eq. (13)
Eq. (14)
1.12 1.14 1.16 1.18
β = 1/T
-8
-6
-4
-2
0
2
4
6
8
lnκ
eq
Eq. (7)
Eq. (16)
simulations
a b
Figure 2. Rate constants and the equilibrium rate. (a) Arrhenius plots
for the forward κ− and the reverse κ+ rate constants. Filled symbols correspond to
numerical results obtained from the stochastic simulations and lines denote the fit to
equations (13) and (14) considering β∗∆F † = 1.16 and β∗ = 1.1489. The parameters
obtained for κ− are A+ = 2.54 × 10−5, ∆E‡+ = −135.2, and γ+ = 5262.6; while for
κ+ are A− = 2.62 × 10−5, ∆E‡− = 130.8, and γ− = 4940.9. By considering τε = 1
and ε = 2, both pre-factors A− and A+ defined by equation (15) yield γ∗ ≈ 2× 10−4.
(b) Arrhenius plot for the equilibrium constant κeq = κ−/k+. Circles correspond to
results that were computed from the numerical data showed in (a). Straight (black) line
denote the linear regression using equation (7), from where one have ∆E = 266 and
∆S = 305.6; while the dashed (red) line correspond to equation (16) by assuming that
∆E‡ = ∆E+ −∆E‡+ = 266 and ∆γ = γ− − γ+ = −321.7. The stochastic simulations
were performed as described in Appendix B and considering the microcanonical entropy
S(E) showed in figure 1 (where it is assumed that kB = 1).
Kinetics of first-order phase transitions from microcanonical thermostatistics 8
enthalpic (∆E) and the entropic (∆S) contributions obtained from equation (7)
should be related, respectively, to the values of ∆E† and ∆S† evaluated from the
microcanonical thermostatistics analysis presented in figure 1(b).
As described in Appendix C, a straightforward way to obtain temperature-
dependent expressions for the rate constants from the microcanonical entropy S(E)
is to evaluate the mean first-passage times from the canonical equilibrium distribution
given by equation (6). Hence, as I discuss below, both τ− and τ+ can be obtained from
the estimates for p(E) at energies close to E−, E+, and E∗, which are obtained from
approximated expressions for the density of states Ω(E). Indeed, by considering the
free-energy profile defined by equation (5), one can estimate Ω(E) directly from the
microcanonical entropy which is given by S(E) = S∗(E) − β∗∆F (E). In particular,
the expansion of the free-energy profile around its maximum at the energy E∗ can be
written as
β∗∆F (E) ≈ β∗∆F † − γ∗
2(E − E∗)2 , (8)
where γ∗ ≈ (db(E)/dE)|E=E∗ is a positive constant. Thus, the canonical probability
distribution around the energy E∗ and at a inverse temperature β close to β∗ can be
estimated as
p∗(E) ≈ Γ∗(β)
Z(β)exp
γ∗2[(E − E∗)− (β − β∗)
γ∗
]2 , (9)
with
Γ∗(β) = exp[S(E±)− β∗E± − β∗∆F † −
((β − β∗) + γ∗E∗)2
2γ∗+γ∗(E∗)2
2
], (10)
where the subscript in E± indicate that either E− or E+ can be used (as long as mixed
notations are avoided). Similarly, by assuming that β∗∆F (E) ≈ (γ±/2)(E−E±)2, with
b(E±) ≈ b(E∗) ≈ β∗ and γ± ≈ (db(E)/dE)|E=E±> 0, the probability distribution close
to the energy E− (or E+), and at β close to β∗, can be approximated by
p±(E) ≈ Γ±(β)
Z(β)exp
−γ±2[(E − E±) +
(β − β∗)γ±
]2 , (11)
with
Γ±(β) = exp[S(E±)− β∗E± +
((β − β∗)− γ±E±)2
2γ±− γ±(E±)2
2
]. (12)
As shown in Appendix C, the forward and the reverse rate constants can be obtained
from equations (9), (10), (11), and (12), and are given, respectively, as
κ− =1
τ−≈ A+ exp
[−∆E‡+(β − β∗)− γ+
2(β − β∗)2
], (13)
κ+ =1
τ+
≈ A− exp[−∆E‡−(β − β∗)− γ−
2(β − β∗)2
], (14)
Kinetics of first-order phase transitions from microcanonical thermostatistics 9
with
A± =
√√√√ ε4γ∗
8πτ 2ε (∆E‡±)2
e−β∗∆F † , (15)
where γ± = (γ±)−1 + (γ∗)−1, ∆E‡± = E∗ − E±, and τε is a characteristic time scale
involved in a microscopic energy transition. Also, from equations (13) and (14), one can
readily evaluate the equilibrium rate constant, which is given by
κeq =κ−κ+
≈ A exp[∆E‡(β − β∗) +
∆γ
2(β − β∗)2
], (16)
where ∆E‡ = ∆E‡− −∆E‡+ = E+ − E−, A = ∆E‡−/|∆E‡+|, and ∆γ = (γ−)−1 − (γ+)−1.
As shown in figure 2(b), equation (16) fits the numerical results obtained from
the stochastic simulations very well and, by assuming that β∗ = 1.1489, one finds
that ∆γ = −321.7, ∆E‡− = 130.8, ∆E‡+ = −135.2, so that ∆E‡ = 266, which is
the same value obtained for ∆E defined in equation (7). One should note that,
since ∆S‡ = β∗∆E‡, equation (16) would retrieve the usual van’t Hoff’s expression,
equation (7), only if the free-energy profile β∗∆F (E) present symmetric wells, that
is, if |∆E‡+| = |∆E‡−| and ∆γ = 0, which is not the case. Importantly, the equal
height criteria based on the probability distribution p(E) [23] is obtained when β = β∗,
however, for non-symmetric wells, equation (16) yields keq(β∗) = ∆E‡−/|∆E‡+| 6= 1, so
that κ− is slightly different from κ+ at β∗. Alternatively, one may consider the equal
area criteria (see, e.g., reference [29]) where the equality κ− = κ+ (i.e., keq = 1) occur
at a temperature Tm = 1/βm, with βm = β∗ + δ and δ ≈ (∆E‡)−1 ln(|∆E‡+|/∆E‡−
).
It is worth mentioning that, when δ β∗, the ratios ∆S‡/∆E‡ and ∆S/∆E will
give the same inverse transition temperature T ∗ evaluated from their microcanonical
counterparts, i.e., T ∗ = 1/β∗ computed from ∆S† and ∆E† as in equation (4).
Even so, one should note that the estimates of the latent heat obtained from the
fit of equations (7) and (16) to the numerical data present slightly smaller values,
i.e., ∆E‡ = ∆E = 266, than the microcanonical latent heat, ∆E† = 288, which was
evaluated directly from S(E), as illustrated in figure 1(b). Table A1 shows that one have
∆E‡ ≈ 0.9 ∆E† also for other microcanonical entropies that were obtained with different
parameters and which display different free-energy barriers β∗∆F †. Thus, I note that
such deviation is not particular of the parameters used to obtain figure 1. In any
case, one should recognize that it is a fair agreement considering all the approximations
involved in the derivation of equation (16), and also that the mean first-passage times
were evaluated between the same energies E− and E+ for all temperatures even though
the maxima of p(E) changed with T .
Figure 2(a) indicates that the numerical results for the forward and the reverse rate
constants obtained from the stochastic simulations are well fitted by expressions (13)
and (14). Importantly, by leaving just γ∗ as a free parameter, that is, by considering
the above values of ∆E‡−, ∆E‡+, with γ− and γ+ consistent with the value ∆γ given
by equation (16), and the free-energy barrier β∗∆F † obtained from the microcanonical
analysis, one have that γ∗ ≈ 2× 10−4, which is consistent with the value one finds from
Kinetics of first-order phase transitions from microcanonical thermostatistics 10
the fit of the peak of the free-energy profile β∗∆F (E) to the Gaussian function defined
by equation (8) (data not shown). That value of γ∗ ensures a quantitative agreement
even for the pre-factors A− and A+, which can be evaluated independently through
equation (15).
In the following I validate my approach by applying the expressions derived in
this section for the rate constants to describe the experimental data on the kinetics of
first-order phase transitions of two molecular systems.
3.2. Protein folding kinetics and protein stability
As a first example, I consider the temperature-dependent kinetics related to protein
folding-unfolding transitions [30], where one can use several experimental techniques [29,
31, 32] to probe the refolding (f) and the unfolding (u) rate constants. Usually,
the rate constants are determined in terms of Arrhenius-like expressions with the
effective free-energies, also known as activation energies, defined as ∆G(u)f (T ) =
−kBT ln[κ(u)f (T )/κ0], so that [31]
κ(u)f (T ) ≈ κ0 exp−[∆H(u)f − T∆S(u)f + ∆C(u)f
p
((T − Tm)− T ln
(T
Tm
))]/kBT
, (17)
where Tm is a reference temperature, e.g., the midpoint of thermal denaturation [33],
∆H(u)f and ∆S(u)f are reference enthalpies and entropies at Tm, respectively, and ∆C(u)fp
are estimates for the changes in heat capacity at constant pressure; κ0 is an effective
kinetic constant that sets the units of the rate constants and, in this case, is related to
the viscosity of the solvent [31]. Alternatively, by assuming that the rate constants can
be expressed by equations (13) and (14), one finds that the refolding (i.e., forward) and
unfolding (i.e., reverse) rate constants can be written as
κ(u)f (T ) ≈ A0 exp
−[∆E‡(u)f
(1− T
Tm
)+
γ(u)f
2kBT
(1− T
Tm
)2]/
kBT
, (18)
where the parameters ∆E‡(u)f and γ(u)f are related, respectively, to the energy differences
and curvatures that are obtained from the microcanonical free-energy profiles (see
section 3.1).
As indicated by the results presented in figure 3(a), the expressions (17) and (18)
can describe equally well the temperature dependence of the rate constants obtained
from experiments on the folding-unfolding transition of the FRET-PGK protein [31]. By
considering the values of the parameters obtained from the data displayed in figure 3(a),
one can readily identify that the first two terms in the exponential of equation (17),
which are related to the enthalpic and entropic contributions, should correspond to
the first term in the exponential of equation (18), so that ∆H(u)f ≈ ∆E‡(u)f and
∆S(u)f ≈ ∆S‡(u)f = ∆E‡(u)f/Tm. Importantly, despite the clear difference between the
last terms in the exponentials of equations (17) and (18), one can verify from the data
presented in figure 3(a) that the ratio between those terms is almost constant over
the range of temperatures considered (data not shown). Indeed, by assuming that
Kinetics of first-order phase transitions from microcanonical thermostatistics 11
310 315 320
T (K)
-8
-6
-4
-2
0
ln[κ
(T )
/ s
-1]
κf
κu T
m
Eq. (17)
Eq. (18)
exp. data from
reference [30]
280 300 320 340
T (K)
-10
0
10
20
30
∆G
(T )
(
kJ/
mol)
native denatured
Tm
exp. data from ref. [28]
Eq. (19)
Eq. (20)
a b
Figure 3. Refolding and unfolding rates and the protein folding stability.
(a) Filled symbols correspond to the experimental rate constants extracted from
reference [31] for FRET-PGK protein. Dashed (red) lines correspond to equation (17)
with Tm = 313.05 K (i.e., Tm = 39.9oC) and κ0 = 0.24 s−1 for both refolding (κf )
and unfolding (κu) rate constants; for the refolding rate constant one have that
∆Hf = −333 kJ/mol, ∆Sf = −1.06 kJ/(mol.K), and ∆Cfp = −45 kJ/(mol.K); while
for the unfolding rate the values are ∆Hu = 345 kJ/mol, ∆Su = 1.10 kJ/(mol.K),
∆Cup = −37.8 kJ/(mol.K). Straight (black) lines correspond to equation (18) with
A0 = κ0; for κf one have that ∆E‡f = −334 kJ/mol and γf = 37042 (kJ/mol)2;
while for κu the values are ∆E‡u = 344.6 kJ/mol and γu = 31047 (kJ/mol)2. (b)
Circles correspond to the experimental data for the free-energy ∆G(T ) extracted
from reference [29] for SOD(I35A) enzyme in PBS buffer; the dashed (purple) line
corresponds to equation (19) with Tm = 308.75 K (i.e., Tm = 35.6oC), ∆H =
−144.8 kJ/mol, ∆S = −0.47 kJ/(mol.K), and ∆Cp = −7.76 kJ/(mol.K); and the
continuous (black) line corresponds to equation (20) with ∆E‡ = −145.1 kJ/mol and
∆γ = 5827.3 (kJ/mol)2. Grey areas indicate regions that denatured states becomes
favourable, including those related to cold denaturation [6] at low temperatures.
T is close to the transition temperature, i.e., T ≈ Tm, one can use the approximation
ln(1+x) ≈ x−x2/2 with x ≈ (T−Tm)/Tm, so that a comparison between equations (17)
and (18) indicates that ∆C(u)fp ≈ −γ(u)f/kBT
2m.
It is worth mentioning that both forward and reverse rate constants given by
equations (17) and (18) should be interpreted as pseudo-equilibrium constants [34],
and are, in fact, related to relaxation kinetics of reversible processes close to the
folding-unfolding transition [35] (see Appendix D for details). The actual equilibrium
constant can be estimated experimentally as κeq = [N ]/[D] from measurements of the
concentrations [N] and [D] of proteins (at equilibrium) in their native and denatured
states, respectively (see, e.g., references [31, 32, 29]). Usually, it is assumed that the
equilibrium constant can be written as an Arrhenius-like expression, i.e., κeq = κf/κu =
e−∆G(T )/kBT , with the corresponding effective free-energy given by [29, 36, 37]
∆G(T ) = ∆H − T∆S + ∆Cp
[(T − Tm)− T ln
(T
Tm
)], (19)
where ∆H = ∆Hf −∆Hu, ∆S = ∆Sf −∆Su, and ∆Cp = ∆Cfp −∆Cu
p . Alternatively,
Kinetics of first-order phase transitions from microcanonical thermostatistics 12
the equilibrium constant κeq can be evaluated from the rate constants κf and κu defined
by equation (18), so that the protein folding stability is determined from an effective
free-energy that is given by
∆G(T ) = ∆E‡(
1− T
Tm
)+
∆γ
2kBT
(1− T
Tm
)2
, (20)
where ∆E‡ = ∆E‡f − ∆E‡u and ∆γ = γf − γu. I note that the derivation of the
above equation assumes the equal area criteria, yet, one can consider equation (16) to
obtain a similar free-energy but assuming the equal height criteria (see the discussion
in section 3.1).
Clearly, the results displayed in figure 3(b) indicate that the temperature
dependence of the free-energy ∆G(T ) obtained from experiments on SOD enzyme [29]
can be well described by both equations (19) and (20). Similarly to what is observed
for the rate constants, one can identify that the first term in equation (20) should
correspond to the first two terms in equation (19), which are related to the enthalpic
and entropic contributions, so that ∆H ≈ ∆E‡ and ∆S ≈ ∆S‡ = ∆E‡/T ∗. And, again,
one can verify that the ratio between the last terms in equations (19) and (20) is almost
constant for the data presented in figure 3(b), and, by comparing the approximated
expressions, one finds that ∆Cp ≈ −∆γ/kBT2m. In any case, one should note that the
derivations of equations (17) and (19) consider that ∆Cp and ∆C(u)fp are independent of
the temperature [37], and assume some approximations which are specific for a particular
protein model [36, 33]. Since the expressions (18) and (20) do not rely neither on that
assumption or on any particular model, they might be useful to provide an interpretation
to inconsistent results related to the protein folding-unfolding transition [6].
In the context of computer simulations of protein folding phenomena, it is worth
mentioning that, instead of the so-called rugged free-energy landscapes [38], the free-
energy profiles β∗∆F (E) that were obtained from replica exchange Monte Carlo
simulations of heteropolymers display rather smooth curves [13], i.e., similar to what
is presented in figure 1(b). One should also note that, although the microcanonical
entropies S(E) present a convex intruder region for systems which present first-order
phase transitions (see, e.g., references [39, 40, 41]), they are somewhat different from the
rugged funnel-like picture commonly used to describe the folding-unfolding transitions,
specially if one recall that, at finite temperatures, the native state of a protein does not
necessarily corresponds to its “ground-state”.
Next I consider a problem that shares some features with the folding-unfolding
transitions [30], which is the nucleation phenomenon.
3.3. Ice nucleation in supercooled water droplets
As discussed in references [42, 43], experiments that measure nucleation rates of ice
in supercooled water droplets at conditions similar to atmospheric cloud formation are
very important to climate science. The state-of-the-art technique involve an ensemble
with M , that is, thousands of micrometre-sized liquid droplets in contact with a cooling
Kinetics of first-order phase transitions from microcanonical thermostatistics 13
stage that works as a thermal reservoir. Hence, by assuming that, at a time t′, nL(t′)
droplets are in the liquid state at a temperature T ′ = T (t′), one can relate the number
of droplets ∆nF that should freeze after a interval of time ∆t = t′′− t′ to the nucleation
rate coefficient J(T ′) through an analytical expression that is given by [42, 44, 45]
∆nF (T ′) = nF (t′′)− nF (t′) = nL(t′)(1− e−J(T ′)V∆t) , (21)
where V is the volume attributed to a single liquid droplet. In practice, the experiments
are done at a constant cooling rate, e.g., r = −1 K/min, and one measures the fraction
of frozen droplets fF (t) = nF (t)/M at consecutive (discrete) times, i.e., t′′ = t′ + ∆t.
Since one can associate an average temperature T ′ to a given time interval [t′, t′ + ∆t[,
equation (21) can be inverted to provide an experimental estimate for the nucleation
rate coefficient, that is [44],
J(T ′) = − 1
V∆tln
[1− fF (t′′)
1− fF (t′)
]. (22)
Importantly, one should note that the stochastic description used to obtain
equations (21) and (22) assumes that the whole droplet freezes as a consequence of
a single microscopic nucleation event [43, 44, 45]. This means that the nucleation rate
j(T ), which is usually defined by an Arrhenius-like expression just like equation (1),
can be directly related to the nucleation rate coefficient as j(T ) = J(T )V [44]. Also, it
turns out that the rate theory developed in section 3.1 is very convenient to describe
those ice nucleation experiments, since the expressions for the rate constants derived in
that section correspond to the phase transformation kinetics of the whole system, even
though “whole” in this case (just as in computational simulations) means a very small
portion of a micrometre-sized droplet.
Because the nucleation experiments are performed with a varying temperature, one
must consider that the number of frozen droplets nF (t) at a time t might not correspond
to its expected equilibrium value. Indeed, the time evolution of nF (t) should be governed
by a relaxation kinetics according to equation (D.6), with an effective relaxation rate
given by Jobs, just as discussed in Appendix D. Hence, the number of droplets ∆nF that
will freeze after a interval of time ∆t at a temperature T will be approximately given by
equation (D.10), which is identical to equation (21), from where one can identify that
J(T )V ≈ J−(T )V = κ−(T ), with the forward rate constant κ− given by equation (13).
Thus, by considering equation (D.9), one can write the nucleation rate coefficient as a
function of the temperature as
J(T ) ≈ J0 exp
−(
1
T− 1
T ∗
) [∆E‡+kB
+γ+
2k2B
(1
T− 1
T ∗
)](23)
where
J0 ≈Dε
V |∆E‡+|
√γ∗
2πe−β
∗∆F † , (24)
with the parameters defined as in section 3.1.
Kinetics of first-order phase transitions from microcanonical thermostatistics 14
0.86 0.88 0.9 0.92
T
0.1
0.2
0.3
0.4
f F (T
)
235 236 237
T (K)
0.1
0.2
0.3
0.4
f F (T
)
235 236
T (K)
106
107
108
109
J(T
) (
cm-3s-1
)
Eq. (25)
Eq. (26)
0.86 0.88T
10-6
10-5
10-4
J(T
)Eq. (23)
Eq. (26)
a bsimulations experiments
Figure 4. Ice nucleation rates in supercooled water droplets. (a) Circles
correspond to the fraction of frozen droplets fF (T ) as a function of the temperature
T obtained numerically with aid of equation (D.6) by considering a relaxation rate
given by Jobs = (κ− + κ+)/V , with the rate constants κ− and κ+ computed with
the parameters obtained from figure 2(a) (it is also assumed that M = 104, V = 1,
∆t = 100, and kB = 1). Inset shows the nucleation rate coefficient: circles correspond
to J(T ) obtained from fF (T ) through equation (22), while the straight (black) and
dashed (red) lines are given by equations (23) and (26) with the same parameters used
to generate the data in the main panel, i.e., A+ = 2.54 × 10−5, ∆E‡+ = −135.2,
and γ+ = 5262.6. (b) Circles correspond to experimental data on ice nucleation
in micrometre-sized water droplets extracted from reference [45]. The main panel
shows the fraction of frozen droplets fF (T ) while the inset include the nucleation
rate coefficient J(T ) evaluated from fF (T ) through equation (22). In the inset the
straight (black) line corresponds to the fit to equation (26) by assuming T ∗ = 273.15 K
and ∆E‡+/kB(T ∗)2 = −3.9126 K−1, which yields γ+/(2[kB(T ∗)2]2) ≈ −0.01 K−2 and
J0 ≈ 10−59 cm−3.s−1, while the dashed (blue) line corresponds to equation (25) with
a = −3.9126 K−1 and b = 939.916 (see reference [45] for details).
Figure 4(a) illustrates what one usually observes from ice nucleation experiments
(see, e.g., figure 4(b)) and include numerical results that were produced in order to
validate equation (23). The main panel shows the fraction of replicas of the system that
are in the low energy phase (e.g., frozen droplets) obtained with aid of equation (D.6)
by considering Jobs = (κ−+κ+)/V , with the rate constants given by the data displayed
in figure 2(a) and arbitrary values of V and ∆t. As shown in the inset of figure 4(a), the
nucleation rate J(T ) given by equation (23) with the same parameters of the forward
rate κ− displayed in figure 2(a) is very similar to the numerical estimates evaluated from
fF (T ) through equation (22). Such result corroborates the idea discussed in Appendix
D that, indeed, the nucleation rate coefficient can be well approximated by the forward
rate constant, that is, J(T ) ≈ Jobs(T ) ≈ J−(T ). Interestingly, that result also justifies
the use of the forward rate, i.e., J− = κ−/V , as an estimate for the nucleation rate used
in numerical simulations [46].
In the context of atmospheric ice formation, one often resort to empirical
approaches [42]. The simplest phenomenological expression which is used by
Kinetics of first-order phase transitions from microcanonical thermostatistics 15
experimentalists to describe the nucleation rate coefficient is given by [44, 45]
ln J(T ) = aT + b , (25)
with a and b defined as empirical parameters. As shown in figure 4(b), which includes
the experimental data on homogeneous nucleation of ice in supercooled water droplets
extracted from reference [45], equation (25) with a = −3.9126 K−1 and b = 939.916
describes the data for J(T ) well (see the inset of that figure). In order to provide
some reasoning behind that phenomenological expression, one can assume that the
temperature of the thermal reservoir is not too far from the transition temperature,
i.e., T ≈ T ∗, so that the logarithm of equation (23) can be approximately rewritten as
ln J(T ) ≈ ln J0 + (T − T ∗)[
∆E‡+kB(T ∗)2
− γ+
2[kB(T ∗)2]2(T − T ∗)
], (26)
where the parameters are the same as in equation (23). As shown in the inset of
the figure 4(b), equation (26) can be used to fit the experimental equally well as
equation (25). And, if one assumes that ∆E‡+/kB(T ∗)2 is given by the value of a obtained
from the empirical expression, the value of the parameter γ+ determined by fitting the
experimental data to equation (26) is very small (as shown in the inset figure).
Unfortunately, the data available from the most of the experimental studies on
homogeneous nucleation of ice are given only for a narrow range of temperatures
just like in the figure 4(b), thus one cannot conclude (based solely on the fit) that
the obtained parameters are reliable or not. In any case, here I use the numerical
results evaluated from Jobs = (κ− + κ+)/V in order to validate the approximated
expression (26). Remarkably, as shown in the inset of the figure 4(a), equation (26)
can be used to describe the numerical data for J(T ) just as well as equation (23), and,
more importantly, with the same values for the parameters J0, T ∗, ∆E+, and γ+.
The main advantage of evaluate the nucleation rate from microcanonical free-energy
profiles is that it does not require one to define an equimolecular dividing surface that
separates the molecules that are in new phase from the molecules that are still in the
old phase [7, 8]. As mentioned in the introduction, the microcanonical thermostatistics
analysis can be considered a shape-free method, so it should not present any difficulties
related to geometric features of the system that are shared by most of nucleation theories
which are based on the classical nucleation theory. In the context of atmospheric ice
nucleation [42], in particular, equations (23) and (26) should provide experimentalists
an alternative way to analyse their data without having to resort to interfacial energies
which are characterized by an arbitrary power-law dependence on T .
4. Concluding remarks
Although inferences about the kinetics of first-order phase transitions based on
microcanonical free-energy profiles were suggested before in references [12, 13],
microcanonical thermostatistics have been used mainly to describe the equilibrium
properties of molecular systems (see, e.g., references [16, 47, 48, 49, 50, 51]). In
Kinetics of first-order phase transitions from microcanonical thermostatistics 16
this paper I have extended the use of microcanonical thermostatistics in order to
develop a rate theory which provides simple temperature-dependent expressions for
all rate constants, i.e., the forward (κ−) and the reverse (κ+) rate constants given by
equations (13) and (14), respectively, and the equilibrium constant (κeq), which is given
by equation (16). Those expressions were validated through numerical results obtained
from stochastic simulations, and I showed that κ− and κ+ can display non-Arrhenius
behaviours (see figure 2(a)), just as it is observed experimentally for the kinetics data
on protein folding-unfolding transitions presented in figure 3(a).
Since the rate constants were derived from the mean first-passage times and
those, in turn, were estimated from the microcanonical free-energy profiles β∗∆F (E),
their analytical expressions should be useful in providing the kinetics of finite
molecular systems directly from their entropies S(E), or inverse of the microcanonical
temperatures b(E), which are quantities that can be determined from several
computational algorithms [18, 19, 20, 21, 22]. Importantly, one should note that,
although the microcanonical free-energy profiles used here were evaluated as a function
of the energy E (see, e.g., references [22, 51]), it should be straightforward to generalize
the expressions for the rates in terms of enthalpic-dependent entropies [52, 53].
In addition, it is worth mentioning that in order to implement the stochastic
simulations (see Appendix B) as well as to derive the analytical expressions for the
mean-first passage times (see Appendix C), I have assumed that the stochastic processes
are defined by one-dimensional Markov chains with an energy-independent diffusion
coefficient Dε. In order to extend the approach to stochastic processes that are
obtained from more sophisticated computational simulations where the energy E is
a variable projected from a high-dimensional space, one might have to consider an
energy-dependent diffusion coefficient D(E) (see references [54, 55]). Nevertheless, if
the diffusion coefficient D(E) present a deep valley close to the transition energy E∗
(see, e.g., [28]), Dε might be effectively replaced by D(E∗). Alternatively, one can
pursuit an approach that involve a temperature-dependent diffusion coefficient D(T )
(see, e.g., reference [56]), but that is beyond the scope of the present work.
Finally, one should note that, because the microcanonical thermostatistics analysis
is a shape-free method, the temperature-dependent expression for the nucleation rate
coefficient J(T ), i.e., equation (23) derived from the free-energy profiles, can be used
to describe both homogeneous and heterogeneous nucleation processes. Also, further
applications of the nucleation rate coefficients presented in section 3.3 might include
experiments that involve temperature-dependent kinetics such as the time-temperature-
transformation diagrams of glass-based materials, e.g., glass-ceramics and metallic
glasses [57], as well as the aggregation kinetics of biomolecules [1, 2, 58, 59].
Appendix A. Microcanonical model for phase transitions
Simplified models have been largely used in the study of phase transitions [60, 61, 62, 63].
Here I introduce an effective model to obtain S(E) by assuming that its microcanonical
Kinetics of first-order phase transitions from microcanonical thermostatistics 17
temperature is given by
T (E) = −a0E + a1eb1E − a2e
−b2E + a3 . (A.1)
This function is used to evaluate the inverse of the microcanonical temperature through
the relation b(E) = 1/T (E). Importantly, both first-order and continuous phase
transitions can be modelled by equation (A.1). First-order phase transitions are
characterized by S-shaped microcanonical temperatures [13, 16, 39, 64] which can be
obtained when the positive function b(E) present an inflexion point at a positive energy
value E∗. This energy can be found through the condition d2T (E)/dE2 = 0, which
leads to E∗ = −(b1 + b2)−1 ln [a1b21/a2b
22] or, equivalently, to the condition a1b
21 < a2b
22.
The curve b(E) presented in figure 1(a) was obtained with a0 = 0.0011, a1 = 0.02,
b1 = 0.005, a2 = 1.2, b2 = 0.01, and a3 = 1.18. Note that by choosing a3 = a2 − a1
one have that “ground-state” energy is reached at zero absolute temperature, that is,
T (0) = 0. For those parameters, the transition temperature is T ∗ = T (E∗) = 0.8704
with β∗ = b(E∗) = 1/T ∗ = 1.1489. Table A1 list the parameters that can be used to
generate different microcanonical temperatures that display S-shaped curves and which
can be used to describe first-order phase transitions. In order to perform the stochastic
simulations explained below, I consider a energy discretization Em = E0 + mε, with
E0 = 0, ε = 2, and m = 0, 1, . . . . Hence, the microcanonical entropy is estimated
from the values of b(Em) using a piece-wise relation S(Em) = b(Em)Em− a(Em), where
the values of a(Em) are determined from b(Em) through the recurrence relations of the
multicanonical algorithm [18, 22, 51].
Table A1. Parameters used to generate the different microcanonical temperatures
T (E) defined by equation (A.1); inverse of the microcanonical transition temperature,
β∗; free-energy barrier, β∗∆F †; microcanonical latent heat, ∆E†; latent heat obtained
from equation (16), ∆E‡;
Parameters β∗ β∗∆F † ∆E† ∆E‡
a0 = 0.0011, a1 = 0.02, b1 = 0.005, a2 = 1.2, b2 = 0.01 1.1489 1.16 288 266
a0 = 0.0009, a1 = 0.02, b1 = 0.0048, a2 = 1.12, b2 = 0.012 1.1441 1.53 320 286
a0 = 0.001, a1 = 0.02, b1 = 0.0049, a2 = 1.2, b2 = 0.012 1.0859 2.15 350 316
a0 = 0.001, a1 = 0.02, b1 = 0.00495, a2 = 1.2, b2 = 0.013 1.0747 2.74 370 338
a0 = 0.0013, a1 = 0.02, b1 = 0.0053, a2 = 1.27, b2 = 0.011 1.1180 3.30 360 330
Appendix B. Stochastic simulations
From the microcanonical entropy S(Em) at a discretized value of energy Em, one can
determine the density of states Ω(Em) = eS(Em) and evaluate the equilibrium canonical
distribution as p(Em) = [Z(β)]−1Ω(Em)e−βEm . It is worth mentioning that special care
is needed in the evaluation of the partition function Z(β), since it requires the large
number summation technique that is described in reference [18]. Also, in order to avoid
numerical instabilities, a threshold value of pmin = 10−10 is used to set the range where
Kinetics of first-order phase transitions from microcanonical thermostatistics 18
1×105
1.5×105
2×105
steps
200
400
600
ener
gy
τ-
τ+
H(E )
p(E )
E-
E+
histogram
Figure B1. Markov chain and energy histogram obtained from stochastic
simulations. Left panel include the time series produced by the stochastic protocol
described in Appendix B for the microcanonical entropy S(E) displayed in figure 1(b),
for a temperature T = 0.875, which is just above the transition temperature T ∗ =
0.8704 (here it is assumed that kB = 1). Horizontal dashed (grey) lines indicate the
energies E− = 232 and E+ = 520 which are used to evaluate the first-passage times,
which are denoted by τ− and τ+ (see reference [28] for details). Right panel shows
the energy histogram H(E) obtained from the time series with Ns = 3 × 109 steps,
while the straight (black) line correspond to the canonical distribution p(E) given by
Eq. (6).
p(E) > 0, that is defined between the energies Ei (initial) and Ef (final). Those values
of energy are determined from the probabilities p(Ei) and p(Ef ) which are just above
the threshold value pmin. The kinetics of the system is simulated from microscopic
transitions using a simple stochastic approach, where the system with energy Em go to
a energy Em+1 with probability Tm,m+1 or to a energy Em−1 with probability Tm,m−1.
The transition probabilities Tm,n define a stochastic matrix and are obtained from the
equilibrium distribution pm = p(Em). First, the transition probabilities are defined
for m = i, that is, Ti,i = Ω(Ei)/[Ω(Ei) + Ω(Ei+1)] and Ti,i+1 = 1 − Ti,i. Then one
should consider the detailed balance condition, so that Ti+1,i = (pi/pi+1)Ti,i+1. After
that, the rest of the transition probabilities in the stochastic matrix can be updated by
considering the following steps: (i) Tm+1,m = (pm − pm−1Tm−1,m)/pm+1, which comes
from the equilibrium distribution; (ii) Tm,m+1 = (pm+1/pm)Tm+1,m, which comes from
the detailed balance condition; then, steps (i) and (ii) are repeated from m = i + 1 to
m = f − 1. Finally, when m = f one have Tf+1,f = 0 so that Tf,f = 1 − Tf,f−1. Each
stochastic simulation at a given temperature corresponds to Ns = 3× 109 steps starting
at a random energy within the interval [Ei, Ef ]. As shown in Fig. B1, the Markov chain
produced by such stochastic protocol lead to a energy histogram H(E) that is equivalent
to the canonical distribution p(E) given by Eq. (6).
Kinetics of first-order phase transitions from microcanonical thermostatistics 19
Appendix C. Mean first-passage times
As mentioned in section 3.1, the mean first-passage times τ+ and τ− are numerically
evaluated from the stochastic simulations by considering the method of labelled walkers
described in reference [28] (see figure B1 for details). In order to obtain their analytical
expressions, I consider the approach described in reference [54], where the mean-first
passage time for the system to go from E+ to E− is given by
τ− =1
D
∫ E+
E−
dE
p(E)
∫ E+
Ep(E ′)dE ′ , (C.1)
where D = ε2/2τε is an energy-independent diffusion coefficient, ε is the bin size, and
τε determines the time scale involved in a microscopic energy transition. By following a
similar approach discussed in reference [65], one can conveniently evaluate those integrals
as
τ− ≈2τεε2
∫ ∞−∞
dE
p∗(E)
∫ E+
E∗p+(E ′)dE ′ ≈
√√√√8πτ 2ε (∆E‡+)2
ε4γ∗Γ+(β)
Γ∗(β), (C.2)
where I assume that the error function can be approximated by erf(z) ≈ z, and p∗(E)
and p+(E) are given by equations (9) and (11), respectively. Similarly, the mean first-
passage time for the system to go from E− to E+ is given by
τ+ ≈
√√√√8πτ 2ε (∆E‡−)2
ε4γ∗Γ−(β)
Γ∗(β), (C.3)
with Γ∗(β) and Γ−(β) given by equations (10) and (12), respectively.
Appendix D. Relaxation rate and the nucleation rate coefficient
Consider an ensemble with M replicas of the system in contact with a thermal reservoir
at a temperature T . Each replica has a fixed volume V and the system can be either in
a (thermodynamic) phase “+” or in a phase “−”, each which might denote, respectively,
denatured and native states of a protein (if the system corresponds to an isolated
protein in solution just as exemplified in section 3.2), or, liquid and frozen phases (if the
system corresponds to a water droplet, as in the experiments discussed in section 3.3).
By assuming that the interconversion between the two thermodynamic phases can be
described by a reversible reaction of the kind X+ X−, with the forward (“+”→ “−”)
and the reverse (“−” → “+”) rate constants given by [46] J− = κ−/V and J+ = κ+/V ,
respectively, one have that the fraction f−(t) of replicas in the phase “−” changes in
time according to the following flux balance equation [3, 65]
df−(t)
dt= −J+V f−(t) + J−V f+(t) . (D.1)
If the ensemble have a fraction f−(t0) of the replicas in the phase “−” at a time t0, after
a interval of time ∆t this fraction will be given by
f−(t0 + ∆t) = f−(t0) e−JobsV∆t + f eq− (1− e−JobsV∆t) (D.2)
Kinetics of first-order phase transitions from microcanonical thermostatistics 20
where
Jobs = J− + J+ , (D.3)
and
f eq− =
J−Jobs
=κeq
1 + κeq
(D.4)
is the equilibrium fraction of replicas in the phase “−”, with the equilibrium constant
given by κeq = J−/J+.
At equilibrium (and also for stationary states), one can verify through
equation (D.2) that if f−(t0) ≈ f eq− then f−(t0 + ∆t) ≈ f−(t0), which means that
df−/dt ≈ 0. However, if f−(t0 + ∆t) 6= f eq− one can use equation (D.2) to estimate the
change in number of replicas in the phase “−” after a time ∆t as
∆n−(∆t) = M [f−(t0 + ∆t)− f−(t0)] = [neq− − n−(t0)] (1− e−JobsV∆t) , (D.5)
where neq− = Mf eq
− and n−(t0) = Mf−(t0). Since, at any time, n−+ n+ = M , the above
equation can be rewritten as
∆n−(∆t) = n+(t0)
[1− neq
+
n+(t0)
](1− e−JobsV∆t) , (D.6)
where
neq+ = Mf eq
+ =M
1 + κeq
(D.7)
is the equilibrium number of replicas in the phase “+”.
Note that, if n+(t0) is different from neq+ , one can see from equation (D.6) that
∆n−(∆) will be not zero and, in that case, the fraction of replicas in the phase “−”
will relax to equilibrium at a rate Jobs according to equation (D.2). The relaxation rate
Jobs, also known as the rate coefficient, is the main quantity measured in experiments
on relaxation kinetics, e.g., ice nucleation [42] and protein folding [31] experiments.
In practice, one might consider that the temperature of the thermal reservoir which
the replicas are immersed is very low in comparison to the transition temperature (e.g., a
liquid droplet in the supercooled region), so that
ln Jobs = ln (J− + J+) = ln[J−(1 + κ−1
eq
)]≈ ln J− , (D.8)
where it is assumed that ln(1 + κ−1eq ) ≈ 0, since the equilibrium constant κeq 1
for temperatures below the transition temperature T ∗, i.e., β > β∗. In particular, for
the case of microcanonical thermostatistics discussed in section 3.1, one can obtain the
relaxation rate constant Jobs ≈ J− by considering the forward rate constant, κ− = J−V ,
which is given by equation (13), so that
ln Jobs ≈ ln J− ≈ ln J0 − (β − β∗)[∆E‡+ +
γ+
2(β − β∗)
](D.9)
with J0 defined as in equation (24). In addition, at the supercooled region, i.e., T < T ∗,
one have from equation (D.7) that the (expected) equilibrium number of replicas in
Kinetics of first-order phase transitions from microcanonical thermostatistics 21
the phase with higher energy will be small, i.e., neq+ M , hence one can approximate
equation (D.6) to
∆n−(∆t) ≈ n+(t0) (1− e−J−V∆t) , (D.10)
which is an expression that is identical to equation (21). As discussed in section 3.3,
equation (21) can be used to determine the nucleation rate coefficient J(T ) from
experiments on ice nucleation in supercooled water droplets [42]. By comparing
equations (D.10) and (21) one can readily identify that the nucleation rate can be
approximated by the forward rate constant, that is, j(T ) = J(T )V ≈ J−(T )V = κ−(T ).
Acknowledgements
I am especially grateful to Professors Dimo Kashchiev, Nelson Alves, and Erich Meyer,
for the inestimable knowledge they shared with me. I also thank the financial support
of the Brazilian agencies CNPq (Grants No 306302/2018-7 and No 426570/2018-9) and
FAPEMIG (Process APQ-02783-18), although no funding from FAPEMIG was released
until the submission of the present work.
References
[1] Knowles T P J, Vendruscolo M and Dobson C M 2014 Nat. Rev. Mol. Cell Biol. 15 384
[2] Vekilov P G 2016 Progress in Crystal Growth and Characterization of Materials 62 136
[3] Zhou H X 2010 Q. Rev. Biophys. 43 219
[4] Eyring H 1935 J. Chem. Phys. 3 107
[5] Kramers H A 1940 Physica VII 4 284
[6] Cooper A 2010 J. Phys. Chem. Lett., 1 3298
[7] Kashchiev D 2003 Nucleation: Basic Theory with Applications (Butterworth-Heinemann)
[8] Schmelzer J W P 2005 Nucleation Theory and Applications (Wiley-VCH)
[9] Cabriolu R, Kashchiev D and Auer S 2012 J. Chem. Phys. 137 204903
[10] Bingham R J, Rizzi L G, Cabriolu R and Auer S 2013 J Chem Phys. 139 241101
[11] Gross D H E 2001 Microcanonical Thermodynamics (World Scientific)
[12] Zierenberg J, Schierz P and Janke W 2017 Nat. Commun. 8 14546
[13] Frigori R B, Rizzi L G and Alves N A 2013 J. Chem. Phys. 138 015102
[14] Truhlar D G and Kohen A 2001 Proc. Natl. Acad. Sci. USA 98 848
[15] Meyer E 1986 J. Cryst. Growth 74 425
[16] Schnabel S, Seaton D T, Landau D P and Bachmann M 2011 Phys. Rev. E 84 011127
[17] Frenkel D and Smit B 2002 Understanding Molecular Simulation (Academic Press)
[18] Berg B A 2003 Comput. Phys. Commun. 153 397
[19] de Oliveira P M C, Penna T J and Herrmann H J 1996 Braz. J. Phys. 26 677
[20] Wang F and Landau D P 2001 Phys. Rev. Lett. 86 2050
[21] Kim J, Keyes T and Straub J E 2011 J. Chem. Phys. 135 061103
[22] Rizzi L G and Alves N A 2011 J. Chem. Phys. 135 141101
[23] Janke W 1998 Nuclear Physics B 63 631
[24] Lee J and and Kosterlitz J M 1990 Phys. Rev. Lett. 65 137
[25] Crooks G E 2000 Phys. Rev. E 61 2361
[26] Krivov S V 2013 Phys. Rev. E 88 062131
[27] Reimann P, Schmid G J and Hanggi P 1999 Phys. Rev. E 60 R1
[28] Trebst S, Huse D A and Troyer M 2004 Phys. Rev. E 70 70 046701
Kinetics of first-order phase transitions from microcanonical thermostatistics 22
[29] Danielsson J, Mu X, Lang L, Wang H, Binolfi A, Theillet F X, Bekei B, Logan D T, Selenko P,
Wennerstrom H and Oliveberg M 2015 Proc. Natl. Acad. Sci. USA 112 12402
[30] Fersht A 1999 Structure and Mechanism in Protein Science (W. H. Freeman and Company)
[31] Guo M, Xu Y and Gruebele M 2012 Proc. Natl. Acad. Sci. USA 109 17863
[32] Schuler B and Hofmann H 2013 Curr. Opin. Struct. Biol. 23 36
[33] Jackson S E and Fersht A R 1991 Biochemistry 30 10428
[34] Tanford C 1968 Adv. Protein Chem. 23 121
[35] Zwanzig R 1997 Proc. Natl. Acad. Sci. USA 94 148
[36] Baldwin R L 1986 Proc. Natl. Acad. Sci. USA 83 8069
[37] Oliveberg M, Tan Y J and Fersht A R 1995 Proc. Natl. Acad. Sci. USA 92 8926
[38] Janke W 2008 Rugged Free Energy Landscapes. Lect. Notes Phys. 736 (Springer)
[39] Barre J, Mukamel D and Ruffo S 2001 Phys. Rev. Lett. 87 030601
[40] Frigori R B, Rizzi L G and Alves N A 2010 Eur. Phys. J. B: Cond. Matt. Phys. 75 311
[41] Frigori R B, Rizzi L G and Alves N A 2010 J. Phys.: Conf. Ser. 246 012018
[42] Murray B J, Broadley S L, Wilson T W, Bull S J, Wills R H, Christenson H K and Murray E J
2010 Phys. Chem. Chem. Phys. 12 10380
[43] Murray B J, O’Sullivan D, Atkinson J D and Webb M E 2012 Chem. Soc. Rev. 41 6519
[44] Riechers B, Wittbracht F, Hutten A and Koop T 2013 Phys. Chem. Chem. Phys. 15 5873
[45] Atkinson J D, Murray B J and O’Sullivan D 2016 J. Phys. Chem. A 120 6513
[46] Wedekind J, Reguera D and Strey R 2006 J. Chem. Phys. 125 214505
[47] Junghans C, Bachmann M and Janke W 2006 Phys. Rev. Lett. 97 218103
[48] Moddel M, Janke W and Bachmann M 2010 Phys. Chem. Chem. Phys. 12 11548
[49] Bereau T, Bachmann M and Deserno M 2010 J. Am. Chem. Soc. 132 13129
[50] Church M S, Ferry C E and van Giessen A E 2012 J. Chem. Phys. 136 245102
[51] Alves N A, Morero L D and Rizzi L G 2015 Comput. Phys. Commun. 191 125
[52] Cho W J, Kim J, Lee J, Keyes T, Straub J E and Kim K S 2014 Phys. Rev. Lett. 112 157802
[53] Malolepsza E and Keyes T 2015 J. Chem. Theory Comput. 11 5613
[54] Nadler W and Hansmann U H E 2007 Phys. Rev. E 75 026109
[55] Katzgraber H G, Trebst S, Huse D A and Troyer M 2006 J. Stat. Mech. P03018
[56] Bauer B, Gull E, S T, Troyer M and Huse D A 2010 J. Stat. Mech. P01020
[57] Zanotto E D, Tsuchida J E, Schneider J F and Eckert H 2015 Int. Mater. Rev. 60 376
[58] Rizzi L G and Auer S 2015 J. Phys. Chem. B 119 14631
[59] Trugilho L F and Rizzi L G 2020 J. Phys.: Conf. Ser. 1483 012011
[60] Thirring W 1970 Z. Physik 235 339
[61] Bittner E, Nußbaumer A and Janke W 2008 Phys. Rev. Lett. 101 130603
[62] Fiore C E and da Luz M G E 2013 J. Chem. Phys. 138
[63] Matty M, Lancaster L, Griffin W and Swendsen R H 2017 Physica A 467 474
[64] Rizzi L G and Alves N A 2016 Phys. Rev. Lett. 117 239601
[65] Gardiner C 2004 Handbook of Stochastic Methods 3rd ed (Springer)